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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Proof</dfn> From the previous section, we know that the form of the general solution is given by</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_11.html ./knowl/eq2_11.html">
\begin{equation*}
\label{eq2_11}
y=\frac{\int u(x) q(x) \textrm{d} x+C}{u(x)}, \quad u(x)=e^{\int p(x) \textrm{d} x}.
\end{equation*}
</div>
<p class="continuation">Thus, to show that the solution exists and is valid in <span class="process-math">\(\alpha \le x \le \beta\)</span> is equivalent to show (<a href="" class="xref" data-knowl="./knowl/eq2_11.html" title="Equation 2.1.11">(2.1.11)</a> ) satisfies the ODE in <span class="process-math">\(\alpha \le x \le \beta\text{.}\)</span> We know that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_11.html ./knowl/eq2_11.html">
\begin{equation*}
\begin{aligned}
&amp; p(x)~ \textrm{is continuous} ~\rightarrow ~\int p(x) \textrm{d} x ~\textrm{is differentiable} ~\rightarrow~u(x)~ \textrm{is differentiable},\\
&amp; q(x), u(x) ~\textrm{are continuous}~\rightarrow~\int u(x) q(x) \textrm{d} x ~\textrm{are differentiable}.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus, (<a href="" class="xref" data-knowl="./knowl/eq2_11.html" title="Equation 2.1.11">(2.1.11)</a>) is differentiable. Substituting it into the ODE, after some calculation, it is indeed satisfied. Then we may conclude the solution exists and is valid in <span class="process-math">\((\alpha, \beta)\text{.}\)</span> To show that the solution is unique, we only need to show that under <span class="process-math">\(y(x_0)=y_0\text{.}\)</span> <span class="process-math">\(C\)</span> is uniquely determined.</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_11.html ./knowl/eq2_11.html">
\begin{equation*}
\begin{aligned}
&amp;y(x_0)=y_0~\rightarrow~y_0=\frac{\int u(x) q(x) \textrm{d} x \Big|_{x=x_0}+C}{u(x_0)}
~\rightarrow~ C=y_0 ~u(x_0)-\uuline{\int u(x) q(x) \textrm{d} x \Big|_{x=x_0}},\\
&amp;\qquad \qquad \qquad \qquad ~\rightarrow~C ~\textrm{has unique value}.
\end{aligned}
\end{equation*}
</div>
<span class="incontext"><a href="sec2_2.html#p-24" class="internal">in-context</a></span>
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